I am trying to show, using characteristic functions, that the Gamma distribution with shape parameter $\alpha$ and rate parameter $\beta$ tends to the Normal Distribution (when standardised appropriately).
What I have attempted so far:
The Gamma distribution density, expectation, and variance:
$$ f(x) = \frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x} \quad\quad\quad \mathbb{E}[X] = \frac{\alpha}{\beta} \quad\quad\quad \mathbb{V}[X] = \frac{\alpha}{\beta^2} $$
I then standardise the random variable:
$$ Z = \frac{X-\mu}{\sigma} = \frac{X- \alpha/\beta}{\sqrt{\alpha/\beta^2}} = \frac{X- \alpha/\beta}{\sqrt{\alpha}/\beta} $$
This next step is where I am stuck. I want to compute the characteristic function of $Z$, however I am unable to do the integral.
$$ \varphi_Z(t) = \mathbb{E}[e^{itZ}] = \mathbb{E}\left[\exp\left(it \cdot \frac{X-\alpha/\beta}{\sqrt{\alpha}/\beta}\right) \right] = \int_{\mathbb{R^+}} e^{it \cdot \frac{X-\alpha/\beta}{\sqrt{\alpha}/\beta}} \cdot \frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}dx $$
I believe the end result should be:
$$ \varphi_Z(t) = e^{-it\sqrt{\alpha}} \left(1- \frac{it}{\sqrt{\alpha}}\right)^{-\alpha} $$
If someone could help with the integral that would be greatly appreciated.
Assuming the characteristic function I gave is correct:
We can then take the natural logarithm of the c.f. and use the Taylor series for $\ln(1-x) = -\sum_{k=1}^{\infty}\frac{x^k}{k}$:
$$ \begin{aligned} \ln(\varphi_Z(t)) &= -it\sqrt{\alpha} - \alpha\ln\left( 1 - \frac{it}{\sqrt{\alpha}} \right) \\ &= -it\sqrt{\alpha} - \alpha \left( -\frac{it}{\sqrt{\alpha}} - \frac{1}{2}\left(\frac{it}{\sqrt{\alpha}}\right)^2 - \frac{1}{3}\left(\frac{it}{\sqrt{\alpha}}\right)^3 - \cdots \right) \\ &= -it\sqrt{\alpha} + \alpha \left( \frac{it}{\sqrt{\alpha}} - \frac{t^2}{2 \alpha} + \mathcal{O}(\alpha^{-1}) \right) \\ \end{aligned} $$
Taking the limit as $\alpha \rightarrow \infty$, we arrive at $-\frac{t^2}{2}$, implying that:
$$ \mathbb{E}[e^{itZ}] \rightarrow e^{-\frac{t^2}{2}} $$
Which is the characteristic function of the standard normal distribution.
Is the rest of my working correct ?